HC-Cells, Nilpotent Orbits, Primitive Ideals and Weyl ...lie.math.okstate.edu/~binegar/Research/Paris-2008-beamer.pdf · (ii)the Richardon orbits that contain d(O) David Vogan’s

HC-Cells, Nilpotent Orbits, Primitive Ideals and Weyl GroupRepresentations

B. Binegar

Department of MathematicsOklahoma State UniversityStillwater, OK 74078, USA

Representation Theory of Lie Groups and ApplicationsInstitut Henri Poincaré December, 2008

B. Binegar (Oklahoma State University) HC-Cells, Nilpotent Orbits, Primitive Ideals and Weyl Group Representations Paris 2008 1 / 42

Setting

G : set of real points of a connected, complex algebraic group GC defined over RĜadm = {irr. adm. reps of G}

Objective: Understand the organization of Ĝadm in terms of algebraic invariants.


Setting

(Φ,Π,Φ∨,Π∨) : root datum

GC : a connected, linear, complex algebraic group defined over RG : set of real points of a connected, complex algebraic group GC defined over RĜadm = {irr adm reps of G}

Objective: Understand the organization of Ĝadm in terms of algebraic invariants.

irr adm reps←→ irr (g,K) -modules←→ Langlands parms

First Reduction: Ĝadm,λ = {irr. adm. reps of inf char λ}(w/o loss of generality by Borho-Jantzen-Zuckerman translation principle)

Assumption: λ is assumed to be regular and integral

Approach: W -graph structure of Ĝadm,λ −→ algebraic invariantsImplicit Theme: Atlas software makes these ideas computable.


Blocks of irreducible Harish-Chandra modules

“Under the hood” of the atlas software is a parameterization of Ĝadm,ρ in terms of pairs

(x , y) ∈ K\G/B × K∨\G∨/B∨

(There is also a certain compatibility condition between x and y .)

Definition

A block of representations is set of representations for which the pairs (x , y) range overK\G/B × K∨\G∨/B∨ corresponding to fixed real forms of G and G∨.

Atlas’s representation theoretical computations take place block by block.


Example: the blocks of E8

Below is a table is listing the number of elements in each “block” of E8:

e8 E8 (e7, su (2)) E8 (R)e8 0 0 1E8 (e7, su (2)) 0 3150 73410E8 (R) 1 73410 453060

The total number of equivalences classses irreducible Harish-Chandra modules of the splitform E8 (R) with infinitesimal character ρ is thus

1 + 73410 + 453060 = 526 471

Atlas PoV : if you’re going to look outside a particular block you may as well consider allthe blocks of GC.


Cells of Harish-Chandra modules

Definition

Given two reps x , y in HCλ, we say

x y ⇐⇒ ∃ f.d. rep F ⊂∞⊕n=0

g⊗n

s.t. x occurs as subquotient of y ⊗ F

x ∼ y if x y and y x

The equivalence classes for the relation ∼ are called cells (of HC-modules).


Blocks and Cells of HC modules: graph-theoretical formulation

Definition

Given x , y ∈ HCλ, we say

x → y =⇒ x occurs in y ⊗ g

The relation “→” gives HCλ the structure of a directed graph.

“ ” ←→ transitive closure of “→ ”cells of reps ←→ strongly connected components of graph

blocks of reps ←→ connected components of graph

The atlas software explicitly computes this digraph structure as a by-product of itscomputation of the KLV -polynomials.


W -graphs

In fact, atlas’s KLV polynomial computations endow HCλ with even more elaborategraph structure.

Definition

Let B be a block of irr HC modules of inf char λ.The W -graph of B is the weighted digraph where:

the vertices are the elements x ∈ Bthere is an edge x → y of multiplicity m between two vertices if

coefficient of q(|x|−|y|−1)/2 in Px,y (q) = m 6= 0

there is assigned to each vertex x a subset τ (x) of the set of simple roots of g, thedescent set of x .


Example: G2

Below is an example of the (annotated) output of wgraph for the big block of G2.

block descent edge vertices,element set multiplicities

0 {} {}1 {2} {(3,1)}2 {1} {(4,1)}3 {1} {(0,1),(1,1),(6,1)}4 {2} {(0,1),(2,1),(5,1)}5 {1} {(4,1),(8,1)}6 {2} {(3,1),(7,1)}7 {1} {(6,1),(11,1)}8 {2} {(5,1),(10,1)}9 {1,2} {(7,1),(8,1)}

10 {1} {(8,1)}11 {2} {(7,1)}


Example cont’d: the W -graph of G2

The W -graph for this block thus looks like

#0{} (LDS)

#1{2}1 // #3{1}

1

::uuuuuuuuu

1oo

1

��

#4{2}

1

ddJJJJJJJJJ

1

��

1 //#2{1}

1oo

#6{2}

1

OO

1

��

#5{1}

1

��

1

OO

#11{2}1 //

7{1}1

oo

1

OO

#8{2}

1

OO

1 // #10{1}1

oo

#9{1,2}

1

ddIIIIIIIII1

::ttttttttt(trivial)

Note that there are four cells: {#0}, {#1,#3,#6,#7,#11}, {#2,#4,#5,#8,#10},and {#9}.


Invariants

Definition

Let V be an irreducible U (g)-module.

Ann(V ) := {X ∈ U(g) | Xv = 0 , ∀ v ∈ V }

is a two-sided ideal in U(g). It is called the primitive ideal in U(g) attached to V .

Fact: Ann(V ) = Ann(V ′) =⇒ inf ch V = inf ch V ′

The correspondenceHCλ → Prim (g)λ : x 7−→ Ann(x)

is often one-to-one, but generally speaking, several-to-one.

⇒ a fairly fine grained-partitioning of HCλ


Nilpotent Orbits

U(g) is naturally filtered according to

Un(g) = {X ∈ U(g) | X = product of ≤ n elements of g}

The graded algebra

gr(U(g)) =∞⊕n=0

Un (g) /Un−1(g)

is well defined, and, in factgr (U (g)) ≈ S (g)

Definition

Let J be a primitive ideal and set

V(J) = {λ ∈ g∗ | φ(λ) = 0 ∀φ ∈ gr(J)}

The affine variety V (J) is called the associated variety of J.


Fundamental Facts:

V (J) is a closed, G -invariant subset of g∗.In fact,

Theorem

V (J) is the Zariski closure of a single nilpotent orbit in g∗

Definition

Let x ∈ HCλ .The nilpotent orbit attached to x is the unique dense orbit Ox inV (Ann(x)).

Lemma

If x , y belong to the same cell of HC-modules then Ox = Oy .

(ass variety doesn’t change after tensoring with a finite-dim rep)

Different cells can share the same nilpotent orbit

rather coarse invariant of HC-modules


Cell Representations

W acts naturally on the Grothendieck group ZHCλ of irr HC modules of inf char λ viathe “coherent continuation representation”

The W -representation carried by a cell is encoded in its W -graph.

The action of a simple reflection on cell rep corresponds to

Tix =

{−x i ∈ τ(x)x +

∑y :i∈τ(y) my→xy i /∈ τ(x)

The W -representation carried by a cell can be computed by evaluating

χC (si · · · sj) = trace (Ti · · ·Tj)

on a representative si · · · sj of each conjugacy class and then decomposing this characterinto a sum of irreducible characters (i.e., brute force is feasible)

Or via branching rules (Jackson-Noel) (spotting occurence of sign reps of Levi subgroups)


Schematic Recap

nilpotent orbits

root datumatlas // HCλ

atlas // G(HCλ)

77ooooooooooo//

''OOOOO

OOOOOO

Prim(g)λ

Weyl group reps


Nilpotent Orbits: Notation / Apparatus

g = Lie (GR)C; h, a CSA for g;∆ = ∆ (h, g), roots of h in g;Π ⊂ ∆, choice of simple roots in ∆;G : adjoint group of g

Ng : nilpotent cone in g (identifying g∗ with g)S ≡ {special nilpotent orbits} (↔ ass varieties of prim ideals of reg int inf char)d : G\Ng → G\Ng : the Spaltenstein-Barbasch-Vogan duality mapd restricts to an involution on image(d) ≡ S ≡ set of special nilpotent orbits.


Standard Levis and Richardson orbits

Definition

Let Γ be a subset of the simple roots. The corresponding standard Levi subalgebra lΓ isthe subalgebra

lΓ = h +∑α∈ZΓ

gα

Definition

Let p = l + n be the Levi decomposition of a parabolic subalgebra of g and let O benilpotent orbit in l.

indgl (O) := unique dense orbit in G · (O + n)

When O = 0l, indgl (O) is called the Richardson orbit corresponding to l (or to Γ ifl = lΓ).

Remark: Induction preserves “special-ness” and trivial orbits are always special =⇒Richardson orbits are always special.

Not every special orbit is Richardson however.


The Spaltenstein-Vogan Criterion

Proposition. (Spaltenstein) A special orbit O is contained in the closure of a Richardsonorbit indlΓ (0) if and only if the (special) W-rep attached to O contains the signrepresentation of WΓ.

Theorem, (Vogan) Suppose C is a cell of H-C modules with associated special nilpotentorbit OC and let lΓ be a (standard) Levi subalgebra of g. Then

OC ⊂ indglΓ (0lΓ ) ⇐⇒ ∃ x ∈ C s.t. Γ ⊂ τ(x)

Upshot: tau invariants of a cell C constrain which Richardson orbit closures can containOCSet

τ(C) ≡ {τ(x) | x ∈ C} = {τ -invariants of reps in C}

Empirical Fact: # distinct τ(C) = # special nilpotent orbits


Problem: A special orbit is not, in general, determined by the Richardson orbits thatcontain it.

Fact: every special orbit O is determined by

(i) the Richardson orbits that contain O(ii) the Richardon orbits that contain d(O)

David Vogan’s Idea: The tau invariants of a cell might tell us which Richardson orbitscontain OC and which Richardson orbits contain d (OC ).


More Apparatus: tau signatures

Fix Ψ : a set of standard Γ’s: a collection of Γ ∈ 2Π such that

Ψ 1 : 1←−−−−→{conj classes of Levi subalgebras} Γ 7→ G · lΓ

(E.g., choose std Γ’s to be first in the lexicographic ordering of their W -conj class)

Partial Order Ψ as follows:

Γ ≤ Γ′ ⇐⇒ indglΓ (0) ⊂ indglΓ′

(0)

Remark: this ordering tends to reverse the ordering by inclusion.

Definition: The tau signature of an H-C cell C is the pair

τsig (C) ≡(min (τ(C) ∩Ψ) , min

(τ∨(C) ∩Ψ

))Here τ∨(C) is the set of Π-complements of tau invariants in C :

τ∨(C) = {Π− τ(x) | x ∈ C}


Tau signatures for special orbits

Definition: Let O be a special orbit. The tau signature of O is the pair (τ (O) , τ∨ (O))where

τ (O) = min{

Γ ∈ Ψ | O ⊂ indglΓ (0lΓ )}

τ∨ (O) = min{

Γ ∈ Ψ | d (O) ⊂ indglΓ (0lΓ )}

The point: we are using pairs of subsets of simple roots to tell us when a Richardsonorbit closure can contain a special orbit (or its dual).

The same kind of pairs tells us when the orbit attached to a cell can be contained inRichardson orbit (or when the dual cell can be contained in the closure of Richardsonorbit).

Corollary (to S-V criterion)

OC = O ⇐⇒ τsig (C) = τsig (O)


Example: Special Orbits of D5 ≈ so(5, 5)C

O[9,1] hh

vv

O[7,3]

rrr KKK44

**

O[52]

LLL::

$$

O[7,13]

sssdd

zz

O[5,3,1,1]

rrr

9999

999 ii

uu

O[42,12]

99

%%

O[33,1]

O[5,15]

��

��

O[32,22]

LLLO

[32,14]

rrr KKK

O[24,12]

LLLO

[3,17]

sssO

[22,16]

O[110]


Richardson Orbits of D5

O[9,1] = R{}hh

vv

O[7,3] = R{1}

hhhhh VVVVVVV44

**

O[52]

= R{1,3}

VVVVV::

$$

O[7,13]

hhhhhhh dd

zz

O[5,3,12]

= R{1,4,5}

hhhhh

MMMMMM

MMMMM ii

uu

O[42,12]

= R{1,2,4}99

%%

O[33,1]

= R{1,2,4,5} O[5,15]

qqqqqq

qqqqq

O[32,22]

VVVVVVV

O[32,14]

= R{1,3,4,5}

hhhhh VVVVV

O[24,12]

= R{1,2,3,4}

VVVVVVV

O[3,17]

= R{2,3,4,5}

hhhhhhh

O[22,16]

O[110]

= R{1,2,3,4,5}


Tau Signatures of Special Orbits of D5

O{};{1,2,3,4,5}hh

vv

O{1};{1,2,3,4},{2,3,4,5}

ggggg VVVVV44

**

O{1,3};{1,2,3,4}

WWWWW::

$$

O{1};{2,3,4,5}

hhhhhdd

zz

O{1,4,5};{2,3,4,5}

ggggg

MMMMMM

MMMMM ii

uu

O{1,2,4};{1,2,3}88

&&

O{1,2,4,5};{1,2,4,5} O{1,4,5};{1,4,5}

qqqqqq

qqqqq

O{1,2,4,5};{1,4,5}

WWWWWO{1,3,4,5};{1,4,5}

ggggg VVVVV

O{1,2,3,4};{1,3}

WWWWWO{2,3,4,5};{1}

hhhhhO{1,2,3,4},{2,3,4,5};{1}

O{1,2,3,4,5};{}


Tau signatures for cells in the big block of SO(5, 5)

• 365 representations with inf. char. ρ in big block• 32 cells in the big blockOutput of extract-cells

// Individual cells.

// cell #0:

0[0]: {}

// cell #1:

0[1]: {2} --> 1,2

1[3]: {1} --> 0

2[5]: {3} --> 0,3,4

3[13]: {5} --> 2

4[14]: {4} --> 2

*

*

*

// cell #29:

0[328]: {1,2,4,5} --> 2,3

1[340]: {2,3,4,5} --> 2

2[358]: {1,3,4,5} --> 0,1

3[364]: {1,2,3} --> 0

// cell #30:

0[353]: {1,2,3,4,5}

// cell #31:

0[357]: {1,2,3,4,5}


cell # tau signature

0 {} , {1,2,3,4,5}

1 {1} , {1,2,3,4}

2 {1} , {2,3,4,5}

3 {1,3} , {1,3,4,5}

* *

* *

* *

28 {2,3,4,5} , {1}

29 {2,3,4,5} , {1}

30 {1,2,3,4,5} , {}

31 {1,2,3,4,5} , {}

Each of these coincides with the tau signature of a particular nilpotent orbit.


Cell-Orbit Correspondences for SO(5, 5)

O[9,1]

O[7,3]

rrr KKK

O[52]

KKKO

[7,13]

tttO

[5,3,12]

sss

8888

888

O[42,12]

O[33,1]

O[5,15]

��

��

O[32,22]

KKKO

[32,14]

sss JJJ

O[24,12]

KKKO

[3,17]

tttO

[22,16]

O[110]

#0

#1

oooooo OOO

OOO

#2, 5

OOOOO

#3, 4

ooooo

#6, 7, 15, 17

ooooo

@@@@

@@@@

@

#8, 9

#10, 12, 13 #11, 21

~~~~

~~~~

~#16, 18

OOOOO

#14, 19, 20, 22

ooooo OOOO

O

#24, 25

OOOOO #23, 28, 29

ooooo

#26, 27

#30, 31


More Generally:

Exceptional Groups: tables by Spaltenstein list induced orbits and Hasse diagrams.

Tau signatures of special orbits can be done by hand.

1. Use Spaltenstein’s tables to figure out which special orbits are Richardson orbits andto identify the std Γ’s corresponding to the corresponding Levi subalgebra.

2. Place the Richardson orbits in the Hasse diagram of special orbits, and then figure outthe Γ parameters of the minimal Richardson orbits that contain a given special orbit andthe minimal Richardson orbits that contain its Spaltenstein dual

Even E8 can be done by hand.

Classical Groups:

Partition classification −→ closure relationsJust need to

which partitions correspond to special orbits (easy recipes in Collingwood-McGovern)

use dominance ordering of partitions to partial order special orbits

use formulas in [C-M] to determine partitions corresponding to Richardson orbits foreach Γ ∈ Ψ. Place these in the Hasse diagram of special orbits and at the same timepartial order Ψ.

Use the partial ordering of Ψ to ascribe tau signatures to cells (employing atlasdata)

match orbit tau sigs to cell tau sigs


Organization of Prim(g)λ

Standard modules and Irr HC-modules arise naturally in the study of ĜR,adm

Verma and Irr HW modules much more convenient family for discussing primitive ideals.

Set

b = h + n : Borel subalgebra of g ρ = 12

∑α∈∆+(g,h) α

Theorem

Let λ ∈ h∗ and let M (λ) denote the Verma module of highest weight λ− ρ; i.e., the leftU (g)-module

M(λ) := U (g)⊗U(b) Cλ−ρThen(i) The Verma module M (λ) has a unique irreducible quotient module L (λ) which is ofhighest weight λ− ρ.(ii) Every irreducible highest weight module is isomorphic to some L(λ).


Duflo’s theorem

Theorem (Duflo)

For w ∈W (g, h) let

Lw = unique irreducible quotient of M(−wρ)

Thenϕ : W → Prim (g)ρ : w → Ann (Lw )

is a surjection.

Parameterizing Prim (g)ρ is tantamount to understanding the fiber of ϕ : W → Prim (g)ρ


Left Cells in W

Definition

Let ∼ be the equivalence relation on W defined by

w ∼ w ′ ⇐⇒ Ann(Lw ) = Ann(Lw′)

The corresponding equivalence classes of elements of W are called left cells in W .

Definition

Let ≈ be the equivalence relation on W defined by

w ≈ w ′ ⇐⇒ OAnn(Lw ) = OAnn(Lw′ )

The corresponding equivalence classes of elements of W are double cells in W .


Connection with Weyl group reps

Definition

Fix a finite-dimensional representation σ of W . The w ∈W such

C 〈W · pw 〉 ≈ σ

comprise the double cell in W corresponding to σ ∈ Ŵ . The representations of W thatarise in this fashion are called special representations of W .

Theorem

If w ,w ′ belong to same double cell corresponding to a special representation σ ∈ Ŵ .Then

Ann (Lw ) and Ann (Lw′) share the same associated variety.

The unique dense orbit in AV (Ann(Lw )) is a special nilpotent orbit O and theW -rep attached O by the Springer correspondence coincides with σ.

Theorem

Let C ⊂W be a double cell and let σ ∈ Ŵ be the assoc (special) W rep. Then

Card {Ann(Lw | w ∈ C} = dimσ


Two pictures at inf char ρ

HW-modules

W {Lw | w ∈ W } same inf char∪ ∪

C : dbl cell {Lw | w ∈ C} same nilpotent orbit∪ ∪

` : left cell {Lw | w ∈ `} same primitive ideal

HC-modules

B : block of HC-modules {πx | x ∈ B} same inf char∪ ∪

C : cell of HC-modules {πx | x ∈ C} same nilpotent orbit∪ ∪? {πx | x ∈?} same primitive ideal


Tau invariants

Lw := L(−wρ) : simple HWM of highest weight −wρ− ρIw := Ann (Lw )

Iwo : unique max ideal (augmentation ideal, annihilator of triv rep)

Ie = unique min PI at inf char ρ (≤ by inclusion)Isα , α ∈ Π : “pen-minimal” ideals

Theorem

The primitive ideals Isα , α ∈ Π, are all distinct from each other and Ie . Any primitiveideal strictly containing Ie contains at least one of the Isα .

Definition

The tau invariant of a primitive ideal I containing Ie is

τ(I ) = {α ∈ Π | Isα ⊂ I}


Connection with W-graphs of HC-modules

Theorem

Let x be an element of a cell C of HC modules and let τ(x) be its descent set (fromW-graph of C). Then

τ(x) = tau-invariant of Ann(x)


A partitioning of HC cells

W -graph of cell: for each element i ∈ C we attach

a vertex v [i ]

a tau invariant τ [i ] = tau invariant of Ann (πi )

a list of edges with multiplicities e [i ] = [(j1,m1) , (j2,md) , . . . , (jk ,mk)]

τ0 subcells:

x ∼τ0 y ⇐⇒ τ(x) = τ(y)

C =∐

[x]0∈C/∼τ0

[x ]0

(Collecting together reps with common assoc variety and common tau invariant)


A partitioning of cells, cont’d

τ1 subcells: Set τ1(x) = {τ(y) | x → y is an edge}

x ∼τ1 y ⇐⇒ τ(x) = τ(y) and τ1(x) = τ1(y)

C =∐

[x]1∈C/∼τ1

[x ]1

τ2 subcells: Set τ2(x) = {τ1(y) | x → y is an edge}

x ∼τ2 y ⇐⇒ τ0(x) = τ0(y), τ1(x) = τ1(y), τ2(x) = τ2(y)

C =∐

[x]2∈C/∼τ2

[x ]2

... etc.

x ∼τi y ⇐⇒ τ0(x) = τ0(y), . . . , τi (x) = τi (y) and

C =∐

[x]i∈C/∼τi

[x ]i

τ∞ subcells: final stable partitioning : C =∐

[x]∞∈C/∼∞

[x ]τ∞


Lemma

The τ∞ partitioning of a cell of HC-modules is compatible with the partitioning of thecell into subcells consisting of representations with the same primitive ideal:

Ann(x) = Ann(y) =⇒ x and y live in same τ∞-subcell.

(follows from well-definedness of Translation Functor for primitive ideals)

Theorem

Let C be any cell in any real form of any exceptional group G. Then the τ∞ partitioningof C coincides precisely with the partitioning of the cell into sets of irr HC modulessharing the same primitive ideal:

x ∼∞ y ⇐⇒ Ann(x) = Ann(y)

proof:

#P∞-subcells = dim special W -rep attached to cell = max#primitive ideals in cell


picture

Cell W-graph

τ(x1)

dd 88

τ(x2)

ff ::

τ(x)

ddII ::uu

zzuu $$IIτ(x3)

zz &&

τ(x4)

xx $$

· · · · · · · · ·

τ(y1)

ff 88

τ(y)

OO

yyss %%KKτ(y2)

zz &&

τ(y3)

xx $$

. . . . ..

τ(z1)

ff 88

oo //

τ(z)

OO

yytt %%JJτ(z2)

oo

zz &&

τ(z3)

xx $$

//


Thank you


Acknowledgements

Thanks also to

The Atlas for Lie Groups TeamJeffrey Adams Alessandra PantanoDan Barbasch Annegret PaulB– Patrick PoloBill Casselman Siddhartha SahiDan Ciubotaru Susana SalamancaFokko du Cloux John StembridgeScott Crofts Peter TrapaTatiana Howard Marc van LeeuwenSteve Jackson David VoganMonty McGovern Wai Ling YeeAlfred Noel Gregg Zuckerman

and the National Science Foundation for support via the Atlas for Lie Groups andRepresentations FRG (DMS 0554278).


τ(x6)

τ(x5) τ(x7)

τ(x1)

eeJJ

BB��τ(x2)

\\::::::\\

τ(x)

eeLL 99rr

yyrr %%LLτ(x3)

yytt

��::

::::

τ(x4)

yyrrr %%KK %%KK

τ(x8) τ(x10)

τ(x9)

· · · · · · · · · τ(y4) τ(y5)

τ(y1)

eeJJ 99tt

τ(y)

OO

yyss %%KKτ(y2)

yytt

��88

888 τ(y3)

��

� %%KKτ(y6) τ(y9)

τ(y7)

. . . . ..

τ(z5)

τ(z4) τ(z1)

OO

oo // τ(z6)

τ(z)

OO

{{vvv ##HHH

τ(z2)

{{vvv

��55

5555

τ(z3)

��

$$HHH

τ(z7) τ(z9)

τ(z8)


SettingBlocks and Cells of HC-modulesInvariantsPrimitive IdealsPrimitive IdealsPrimitive IdealsPrimitive IdealsNilpotent OrbitsCell Representations

RecapNotationAttaching nilpotent orbits to cellsThe Spaltenstein-Vogan CriterionTau signatures for cellsTau signatures for special orbitsExample: SO(5,5)The general casePrim(g)Duflo's TheoremLeft Cells in WLeft Cells in WTwo PicturesTau Invariants of Primitive IdealsA Partitioning of HC CellsAcknowledgements

Documents

HC-Cells, Nilpotent Orbits, Primitive Ideals and Weyl ...lie.math.okstate.edu/~binegar/Research/Paris-2008-beamer.pdf · (ii)the Richardon orbits that contain d(O) David Vogan’s